On Condensation of a Clause

In this paper, we investigate condensation of a clause. First, we extend a substitution graph introduced by Scheffer et al. (1996) to a total matcher graph. Then, we give a correct proof of the relationship between subsumption and the existence of cliques in a total matcher graph. Next, we introduce the concept of width of a clique in a total matcher graph. As a corollary of the above relationship, we show that the minimum condensation of a clause is corresponding to the clique with the minimum width in a total matcher graph. Finally, we design a greedy algorithm of finding condensation of a clause, as the algorithm of finding cliques with as small width as possible from the total matcher graph of a clause.